3.5 Shape coefficient

　

 The value of the shape coefficient depends for the most part on the three dimensional shape of the object concerned and on the relative angle of attack of the wind. The only realistic way to determine shape coefficients is by means of experiments like full-scale tests or wind tunnel tests using scale models. In this way the effect of mutual interference of nearby individual parts of the complete object is included in the coefficient. Unfortunately data are not available for all ships and offshore constructions. In those cases an estimate based on comparable ship types may be useful. The value of the shape coefficient appears to depend on the computational model used also [11]. It is very important to realize that the values of a shape coefficient for these methods may differ substantially. As a result of this indiscriminate comparison of coefficient values from tests or research may result in wrong conclusions. In the offshore industry the complete structure is often sub divided into characteristic parts. Since classification societies have defined shape coefficients for a number of three dimensional shapes, the sum of the individual wind loads can be determined and consequently also the total wind load. A disadvantage of this method is that the effect of mutual interference of nearby individual parts of the complete object is not included in the coefficients.

　

 Shape coefficients for 'normal' ships; With respect to shape coefficients for 'normal' ships (not offshore constructions) we chose Blendermann's data [5] as point of departure for two reasons: these are the most recent coefficients and they are given for about 48 different vessels and floating docks, a number of them being modern ship types. Comparison of the results of calculation methods of Blendermann with other calculation methods for the same vessel leads to the conclusion that, for the greater part of vessels, these results correspond. However a substantial difference was found for a ballasted tanker and an LNG carrier with spherical tanks. Especially in these cases the user of the computer program has to be (and will be) warned!

　

 Shape coefficients for complex structures; For reasons of consistency (the choice of DNV's formula for wind velocity) we also chose to use DNV's shape coefficients.

　

3.6 Mathematical model of Blendermann ('normal' ships)

　

 Blendermann's [4], [5] starting point is the effective dynamic pressure: an effective value for 1/2 ρ V². This dynamic pressure consists of two components: a part (P1) of the mean dynamic pressure over the mean height of the ship (AL/L) and a part (P2) of the dynamic pressure at the mean height (hm) of the ship. See formulae (12) and (13).

　

　

Where:

k = factor [-],

hm = mean height [m], (AL/L)

　

 k is a factor: ≈0.6 for lateral wind loads and ≈0 for longitudinal wind loads. This factor provides insight into the ratio of contribution to the total pressure drop by the components P1 and P2.

　

3.7 Practical methods used before

　

 Shipping industry; Most, if not all, practical approaches used by shipping companies, masters, navigation officers and pilots are based on a simple form of the formula for dynamic pressure; see formula (1). The following examples of formula often used in daily practice are of particular interest:

　

　

　

 The Netherlands' pilots and a lot of their colleagues abroad have used formula (14) for many years. A major shipping company uses formula (15). None of these formulae takes into account the effects of a vertical wind profile, possible differences of the air mass density or lateral wind coefficient, or the phenomenon gust.

　

 Offshore industry; A practical method used in the offshore industry can be found in [12]. The approach is based on formula (1). The complex structure is sub divided into characteristic parts. Shape coefficients as defined by the classification societies are used to determine the individual wind loads on the components. This method takes into account neither possible differences of the air mass density nor the phenomenon gust. The effects of a vertical wind profile are taken into account to a certain extent: contrary to integration with respect to height this method uses a height coefficient for defined height intervals.

　

3.8 Mathematical model of the computer program

　

 Different calculation models and corresponding data are available; for ships these are the methods and data of e.g. Isherwood [13], Blendermann [4], [5], and the method proposed by the Oil Companies International Marine Forum (OCIMF) [6], [7]. For complex (offshore-like) structures these are the methods and data of several classification societies like e.g. Det Norske Veritas (DNV), [10].

　

 Wind loads on ships; With respect to wind loads on ships we chose Blendermann's calculation model as a starting point for two reasons: this was the most recent model and it is consistent with the earlier choice to use Blendermann's coefficients. The final form of the mathematical model which is used in the computer program is based on Blendermann's model but is not the same. After in-depth analysis it appeared to be possible to transform the combination of formulae (12) and (13), to the shape of formula (10). The corresponding shape coefficients had to be transformed in such a way that, although the calculation model differs, the results of calculations are exactly the same.

　

 Wind loads on complex structures; With respect to wind loads on complex structures the methods of several classification societies were studied. It is beyond the scope of this paper to go into detail regarding all of these methods. We chose DNV's method as point of departure for two reasons: this was the most recent model and DNV uses a logarithmic function to define the height dependent wind velocity. For situations at sea we chose to use DNV's calculation formula (11) in combination with formula (2). In this way a vertical wind profile, air mass density and averaging time interval are all taken into account. Calculations are made for separate characteristic components of the object, using the shape coefficients as proposed by DNV. For 'non-exposed harbor' situations a different method had to be chosen since the influence of different roughness lengths is not included in formula (11). Instead the combination of formulae (3), (8) and (9) is used.

　

　

3.9 Example of extreme differences

　

 An example of extreme difference of cross wind loads caused by an accumulation of wrong assumptions is shown in table 1. The ship concerned is a (future-) large container vessel: LOA = 382 [m],lateral area = 13370 [m²], mean height = 35 [m]. The loads were calculated for a 10-minute mean wind velocity of 20.7 [ms^-1]. A gust factor is not applied.

　

　

3.10 Importance in daily practice

　

 From the example shown in table 1 it can be seen that wrong assumptions may lead to a great difference of calculated wind loads and the actual wind loads. The differences of the input data in table 1 are absolutely not inconceivable: in this particular case the difference in loads means the difference of several tugs! Another important reason for caution when calculating wind loads is the height: for high objects the wind velocity near the top of the object may differ considerably from the velocity near sea level. Since the loads are proportional to the square of the wind velocity this may lead to unpleasant surprises. As mentioned in the introduction of this chapter, this has happened in daily practice. An example of one of the pages of the computer program is shown in figure 3, another example has already been shown in figure 1.

　

4.1 Introduction

　

 In order to follow a track plotted on the chart a course to steer must be determined by the navigators and pilots to match the actual track over the ground as tightly as possible with the desired track. The effect of current is usually taken into account using a vector sum. The magnitude and direction of the current-vector can be taken from tidal stream atlases or from actual measurements. For the effect of wind, as well as an estimation based on practical experience, eventual use can also be made of approximations such as those laid down by Isherwood or Blendermann. In cases where both, wind and current, are present extremely complex hydrodynamic effects can in fact arise. The effects of these have, until now, only been possible for navigators and pilots to guess from experience. Path-prediction using a mathematical model of the ship, reliable under all circumstances, could allow for this. Currently this possibility is being researched world-wide. The results are quite definitely optimistic but are not yet of a nature that is completely reliable in practice. Nonetheless the vector sum, used in practice since time immemorial, has up until now produced satisfying results. What is odd, however, is that this method is only applied to straight tracks. In an article published in the 'Journal of Navigation' in 2000 [14] a method was examined to be applied, in combination with a rate of turn indicator, when going round a bend. The effect of wind was not looked at, only the influence of current was examined. The paper at hand will give a brief overview of the method. For more detailed information regarding the theoretical backgrounds the reader should refer to the article in the Journal of Navigation. Since the approach is quite complicated without using a computer program, this method has not yet find it's way to use in daily practice. This may change when the method is implemented in the computer program by means of fast-time simulations. A provisional version of the program has been used already for instructional purposes. At the time of MARSIM 2003 we hope to be in a position to show you the final version which will be implemented in 'Pilot guide'.

　

4.2 Following a bend, methods used before

　

 In the early 1970s, both Swedish and Dutch pilots took the initiative to follow a curved track in a turn, based on a sound theoretical method (Gylden [15], van Hilten et al.[16], van Roon, [17]). The basis of this comprises the mechanics formula for a circular movement (v=ω^*R) and some trigonometric ratios whereby the precise moment of starting the turn (wheel-over-point or wheel-over-line) can be determined. Refinements can be achieved by applying corrections for the position of the radio navigation aid antenna in relation to the ship's pivoting point, the slow increase of the rate of turn (caused by the moment of inertia about the vertical axis) and the presence of current (van Hilten et al.,[16]). from the considerable experience related to this subject amongst pilots, The following uncertainties have come to the surface over the years:

 The conclusion should certainly not be drawn from this summary that the present approach does not make any sense. If regular control is made during the turn, subsequent errors will become increasingly smaller and the subsequent track will be attained perfectly. However, a troublesome point is that, with some methods, a circular arc cannot be followed. The arc is deformed by the direction of the current, its velocity and the ship's speed.

　

4.3 Overview of the new method

　

 First an important note: the angle between the track over the ground and the heading is referred to in this paper as the current drift angle (α) and should not be confused with a local drift angle caused by rate of turn! Figure 4 shows the relevant speed vectors and angles:

　

　

 The formula derived by MPIN and the RNLNC for the exact rate of turn (RoT) required for accurately following a geographically fixed bend is:

　

　

Where:

RoT = rate of turn [rad.s^-1],

Vtr = speed over ground in track [m.s^-1],

R = radius of turn [m],

Vc = current velocity [m.s^-1],

Vw = longitudinal speed according to electromagnetic log (through water) [m.s^-1],

γ = angle between direction of track and direction of heading [rad],

α = current drift angle [rad],

Vw' = vessel's linear acceleration through the water [m.^-2].

 The relationship at (16) is derived between: RoT on the one side and the values Vw, acceleration, track direction, radius of turning circle (constant), velocity of current (constant) and direction of current (constant) on the other. For every point throughout the turn the RoT can thus be determined. A more or less similar relationship was derived for the longitudinal speed of the Doppler log being one of the variables. As a result of the above mentioned relationships it is possible to determine the exact (and so in fact only correct) rate of turn required in cases where it is necessary to follow a circular arc in relation to the earth's surface. This can be achieved both during constant speed through the water and in case of acceleration(of linear speed).